A random sample of 16 observations from a normal distribution yields the values
Find a 95% confidence interval for the variance of the normal distribution.
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Obtain the maximum likelihood estimate of in
based on an independent sample of size n.
There are three urns, urn 1 having 5 white and 3 black balls, urn II with 4 white and 4 black balls and urn III with 3 white and 5 black balls. A fair die is tossed and
i) urn I is chosen if dice shows up 1, 2 or 3,
ii) urn II is chosen if dice shows up 4 or 5,
iii) urn III is chosen if dice shows up 6.
2 balls are drawn at random from the chosen urn. What is the probability that one of the balls drawn is white and the other black?
Prove that
for all m > 0, n > 0.Hence evaluate (10,7) and
.
Use the Neyman-Pearson Lemma to obtain the best critical region for testing =
against
=
>
in the case of a normal population
N(,
), where
is known. Hence find the power of the test.
For the following frequency table obtain the relative frequencies and the cumulative frequencies of the ‘more than’ type.
| Life (in hours)
499.5 – 649.5 | Frequency
5 |
| 649.5 – 799.5 | 20 |
| 799.5 – 949.5 | 40 |
| 949.5 – 1099.5 | 81 |
| 1099.5 – 1249.5 | 57 |
| 1249.5 – 1399.5 | 34 |
| 1399.5 – 1549.5 | 13 |
| Total | 250 |
For a Poisson distributed random variable X, P(X = 4) = P(X = 5).What is the mean and variance of the distribution?
Let X and Y have the continuous joint distribution with p.d.f.
where a > 0.Compute the marginal densities of X and Y. Check whether X and Y are independent.